Discrete-time Nonlinear Control Research Articles

Further results on memory control of nonlinear discrete-time networked control systems with random input delay

This paper is concerned with the problem of memory state feedback stabilization for a class of nonlinear discrete-time networked control systems with partly known probability distribution of input delay. Different from the common assumptions on the delay in the existing literatures, it is assumed that the probabilities of the delay taking values in a finite set are partly known or fully known in advance. In terms of the information about the occurrence probabilities of the delay taking values in the two finite sets, a new stochastic delay model is proposed, where the probability information of the delay is included in the parameter matrices of the transformed system. Based on the new model, a delay-dependent stabilization criterion is derived using a combination of the Lyapunov functional method, convexity of matrix inequalities, and linear matrix inequality technique. Finally, three illustrative examples are provided to show the effectiveness and applicability of the developed theoretical results.

H∞ fault detection for a class of T-S fuzzy model-based nonlinear networked control systems

In this paper, a robust H ∞ fault detection filter is designed for a class of discrete-time nonlinear networked control systems via T-S fuzzy model with multiple bounded state delay and random packet dropout induced by the limited bandwidth of communication networks. Our aim in this paper is to analyze and design a robust H ∞ full-order fault detection filter, such that the filtering error dynamics is asymptotically mean-square stable with a prescribed H ∞ performance level. Sufficient conditions for the existence of the desired filter are presented. Finally, an example is given to illustrate the effectiveness and applicability of the proposed new design method.

Discrete-Time Event-Triggered Control of Nonlinear Wireless Networked Control Systems

This paper investigates the problem of stabilization of nonlinear discrete-time networked control systems (NCSs) with event-triggering communication scheme in the presence of signal transmission delay. A Takagi-Sugeno (T-S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy event-triggered controller for the stabilization of nonlinear discrete-time NCSs. The idea of the event-triggering communication scheme (i.e., a soft computation algorithm) under consideration is that the current sensor data is transmitted only when the current sensor data and the previously transmitted one satisfy a certain state-dependent trigger condition. By taking the signal transmission delay into consideration and using delay system approach, a T-S fuzzy delay system model is established to describe the nonlinear discrete-time NCSs with event-triggering communication scheme. Attention is focused on the design of fuzzy event-triggered controller which ensures asymptotic stability of the closed-loop fuzzy systems. Linear matrix inequality- (LMI-) based conditions are formulated for the existence of admissible fuzzy event-triggered controller. If these conditions are feasible, a desired fuzzy event-triggered controller can be readily constructed. A nonlinear mass-spring-damper mechanical system is presented to demonstrate the effectiveness of the proposed method.

SOS Based Robust <inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$ </tex-math></inline-formula> Fuzzy Dynamic Output Feedback Control of Nonlinear Networked Control Systems

In this paper, a methodology for designing a fuzzy dynamic output feedback controller for discrete-time nonlinear networked control systems is presented where the nonlinear plant is modelled by a Takagi-Sugeno fuzzy model and the network-induced delays by a finite state Markov process. The transition probability matrix for the Markov process is allowed to be partially known, providing a more practical consideration of the real world. Furthermore, the fuzzy controller's membership functions and premise variables are not assumed to be the same as the plant's membership functions and premise variables, that is, the proposed approach can handle the case, when the premise of the plant are not measurable or delayed. The membership functions of the plant and the controller are approximated as polynomial functions, then incorporated into the controller design. Sufficient conditions for the existence of the controller are derived in terms of sum of square inequalities, which are then solved by YALMIP. Finally, a numerical example is used to demonstrate the validity of the proposed methodology.

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.

Measurement feedback disturbance decoupling in discrete-time nonlinear systems

The paper studies the disturbance decoupling problem by the dynamic measurement feedback for discrete-time nonlinear control systems. To address the problem the algebraic approach, called the algebra of functions, is applied, which allows the system description also depend on non-differentiable functions. A necessary and sufficient condition is given in terms of controlled and (h,f)-invariant functions. Also, algorithms are derived, which find invariant functions and the required feedback. The algorithms are implemented in Mathematica software which is made available over the internet.

Stationary policies for lower bounds on the minimum average cost of discrete-time nonlinear control systems

SUMMARY The paper deals with the control problem of discrete-time nonlinear systems. The main contribution of this note is to present conditions that assure the existence of stationary policies that generate lower bounds for the minimal long-run average cost. These lower bounds coincide with the optimal solution when a mild convergence assumption holds. To illustrate the results, the paper presents an application for the simultaneous state-feedback control problem, and the derived strategy is used to design a real-time simultaneous control for two direct current motor devices. The dynamics of these two devices are written in terms of a nonlinear algebraic matrix recurrence, which in turn represents a particular case for our general nonlinear approach. The optimal gain for the corresponding simultaneous state-feedback problem is obtained, and such a gain was implemented in a laboratory testbed to control simultaneously the two direct current motors. Copyright © 2013 John Wiley & Sons, Ltd.

Stability analysis for a class of nonlinear discrete-time control systems subject to disturbances and to actuator saturation

This paper addresses the stability characterisation problem for a class of nonlinear discrete-time control systems subject to actuator saturation and energy bounded disturbance inputs. The considered class of systems covers all nonlinear discrete-time systems that can be modelled by rational difference equations. Based on quadratic and piecewise quadratic Lyapunov functions, conditions based on linear matrix inequalities are proposed to analyse the asymptotic stability (internal stability) and the ℓ2 input-to-state stability (external stability) of the closed-loop system. The proposed conditions are then incorporated into convex optimisation problems to either maximise an estimate of the region of attraction or a bound on the admissible ℓ2 disturbances, and also to obtain an estimate of the system ℓ2-gain for an admissible set of disturbances.

Attainability analysis in the problem of stochastic equilibria synthesis for nonlinear discrete systems

A nonlinear discrete-time control system forced by stochastic disturbances is considered. We study the problem of synthesis of the regulator which stabilizes an equilibrium of the deterministic system and provides required scattering of random states near this equilibrium for the corresponding stochastic system. Our approach is based on the stochastic sensitivity functions technique. The necessary and important part of the examined control problem is an analysis of attainability. For 2D systems, a detailed investigation of attainability domains is given. A parametrical description of the attainability domains for various types of control inputs in a stochastic Henon model is presented. Application of this technique for suppression of noise-induced chaos is demonstrated.

Feedback linearization and lattice theory

The tools of lattice theory are applied to readdress the static state feedback linearization problem for discrete-time nonlinear control systems. Unlike the earlier results that are based on differential geometry, the new tools are also applicable for nonsmooth systems. In case of analytic systems, close connections are established between the new results and those based on differential one-forms. The Mathematica functions have been developed that implement the algorithms/methods from this paper.

An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems

An iterative algorithm, which is called the integrated optimal control and parameter estimation algorithm, is developed for solving a discrete time nonlinear stochastic control problem. It is based on the integration of the principle of model-reality differences and Kalman filtering theory, where the dynamic integrated system optimization and parameter estimation algorithm are used interactively. In this approach, the weighted least-square output residual is included in the cost function by appropriately monitoring the weighted matrix. An improved linear quadratic Gaussian optimal control model, rather than the original optimal control problem, is solved. Subsequently, the model optimum is updated using the adjusted parameters induced by the differences between the real plant and the model used. These updated solutions converge to the true optimum, despite model-reality differences. For illustration, the optimal control of a nonlinear continuous stirred tank reactor problem is considered and solved by using the method proposed.

Stabilization of Discrete-Time Nonlinear Uncertain Systems by Feedback Based on LS Algorithm

This paper is concerned with the use of the least squares (LS) algorithm to design a feedback control law to stabilize a basic class of discrete-time nonlinear uncertain systems. The result shows that if a certain polynomial criterion is satisfied, the system can be stabilized by feedback based on the LS algorithm for Gaussian distributed noise and unknown parameters. This result thus provides an answer to the question of what are the fundamental limitations of the discrete-time adaptive nonlinear control. This issue of feedback capability has been an open problem for more than ten years since it was put forward.

Fault Detection of Networked Control Systems Based on Sliding Mode Observer

This paper is concerned with the network-based fault detection problem for a class of nonlinear discrete-time networked control systems with multiple communication delays and bounded disturbances. First, a sliding mode based nonlinear discrete observer is proposed. Then the sufficient conditions of sliding motion asymptotical stability are derived by means of the linear matrix inequality (LMI) approach on a designed surface. Then a discrete-time sliding-mode fault observer is designed that is capable of guaranteeing the discrete-time sliding-mode reaching condition of the specified sliding surface. Finally, an illustrative example is provided to show the usefulness and effectiveness of the proposed design method.

On Flatness of Discrete-time Nonlinear Systems

The paper addresses the problem of dynamic feedback linearization of discrete-time nonlinear control systems. Analogously to the continuous-time case, necessary and sufficient conditions for flatness property are obtained and showed to be equivalent to previously known results on feedback linearizability by endogenous dynamic feedback. An example is added to illustrate the results.

MULTIPLE VALUED ITERATIVE DYNAMICS MODELS OF NONLINEAR DISCRETE-TIME CONTROL DYNAMICAL SYSTEMS WITH DISTURBANCE

The study of nonlinear discrete-time control dynamical systems with disturbance is an important topic in control theory. In this paper, we concentrate our efforts to multiple valued iterative dynamical systems, which model the nonlinear discrete-time control dynamical systems with disturbance. After establishing the validity of such modeling, we study the invariant set theory of the multiple valued iterative dynamical systems, including the controllability/reachablity problems of the maximal invariant sets.

T–S fuzzy-model-based robust stabilization for a class of nonlinear discrete-time networked control systems

In this paper, the robust stabilization problem is investigated for a class of nonlinear discrete-time networked control systems (NCSs). To study the system stability and facilitate the design of fuzzy controller, Takagi–Sugeno (T–S) fuzzy models are employed to represent the system dynamics of the nonlinear discrete-time NCSs with effects of the approximation errors taken into account, and a unified model of NCSs in the T–S fuzzy model is proposed by modeling the approximation errors as norm-bounded uncertainties in system metrics, where non-ideal network Quality of Services (QoS), such as data dropout and network-induced delay, are coupled in a unified framework. Then, based on the Lyapunov–Krasovskii functional, sufficient conditions are derived for the existence of a fuzzy controller. By these criteria, two approaches to design a fuzzy controller are developed in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are provided to show the effectiveness of the proposed methods.

LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

Limit-cycle-like control for 2-dimensional discrete-time nonlinear control systems and its application to the Hénon map

In this paper, a control method that generates a desired limit-cycle-like behavior for a 2-dimensional discrete-time nonlinear control system is discussed. First, we define some notations and state the problem formulation. Next, a necessary and sufficient condition of existence of a control input that realizes a desired limit-cycle-like behavior is shown. We then derive a control algorithm to solve the problem on generating limit-cycle-like behaviors, and a modification of the algorithm is also shown. Finally, we apply the two types of algorithms to a chaotic system, the Hénon map, in order to indicate the availability of the proposed method. In addition, by using the control method, we also consider a stabilization problem for the Hénon map.

A thermodynamics-based homogeneous charge compression ignition engine model for adaptive nonlinear controller development

Low-temperature combustion modes, such as homogeneous charge compression ignition, represent a promising means to increase the efficiency and to reduce significantly the emissions of internal combustion engines. Implementation and control are difficult, however, owing to the dependence of the combustion event on the chemical kinetics rather than an external trigger. This work describes a nonlinear control-oriented model developed for a single-cylinder homogeneous charge compression ignition engine, which is physically based on a five-state thermodynamic cycle. This model is aimed at capturing the behavior of an engine which utilizes fully vaporized gasoline-type fuels, exhaust gas recirculation, and intake air heating in order to achieve homogeneous charge compression ignition operation. The onset of combustion, which is vital for control, is modeled using an Arrhenius reaction rate expression which relates the combustion timing to both the charge dilution and the temperature. Despite the fact that homogeneous charge compression ignition combustion is indeed fast, it is not perfectly instantaneous and therefore requires some finite amount of time to occur. To account for this phenomenon within the model, a Δθ term is added which shifts the point of instantaneous combustion from the start of combustion to a point of very-high-energy release based on experimental heat release data. The model is validated against experimental data from a single-cylinder compression ignition engine operating under homogeneous charge compression ignition conditions at two different fueling rates. Parameters relevant to control such as the combustion timing, peak in-cylinder pressure, and pressure rise rates from the simulation agree very well with the experiment in both operating conditions. The extension of the model to other fuels is also investigated via the octane index of several different gasoline-type fuels. Since this nonlinear model is developed from a controls perspective, both the output and the state update equations are formulated such that they are functions of only the control inputs and the state variables, therefore making them directly applicable to state-space methods for control. The result is a discrete-time nonlinear control model which provides a platform for developing and validating various nonlinear control strategies.

Realization of discrete-time nonlinear input–output equations: Polynomial approach

The aim of this paper is to solve reduction and realization problems for discrete-time multi-input multi-output nonlinear control systems by applying the theory of non-commutative polynomials. First, the necessary and sufficient reducibility condition is presented in terms of the greatest common left divisor of two polynomial matrices associated with the set of the higher order input–output (i/o) difference equations of the system. The condition also provides a method for system reduction, i.e. for finding the irreducible representation of the set of the i/o equations, being transfer equivalent to the original system representation. Second, to solve the realization problem, a formula is presented for computing the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The polynomial approach addressed in this paper is more direct and requires noticeably less computations than earlier methods represented in terms of subspaces of differential one-forms.